[PPT] - Algebraic (and Diagrammatic) Structures in Quantum Theory Aleks PowerPoint Presentation

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Algebraic (and Diagrammatic) Structures in Quantum Theory

Aleks Kissinger

Institute for Computing and Information Sciences Radboud University Nijmegen

Quantum Software in NL 2017

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Software = components + composition

Effective software comes from understanding relevant

components and how they compose

Effective quantum software will be exactly the same
But now the “components” could mean:

devices physical processes mathematical processes ρ →

i BiρB† i

...each with its own (related) notion of composition

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Process theories

The moral: take processes as primitive
A process is just a box with inputs and outputs:

quicksort

lists lists

cooking

bacon breakfast eggs food

baby

love poo noise

Composition means forming diagrams of processes:

C A D B

g

A

f h

A

A collection of processes that make sense to ‘plug together’ is

called a process theory

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Quantum Picturalism := quantum process theory

The idea: Describe quantum theory entirely in terms of:

f

B A C D

processes connectivity

B A C

g

A

f

D

h

A

interaction

=

Not in terms of:

Hilbert space
self-adjoint operators, unitary transformations
calculations with matrices/complex numbers
....

(though some may be emergent notions)

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PICTURING QUANTUM PROCESSES

A First Course in Quantum Theory and Diagrammatic Reasoning BOB COECKE ALEKS KISSINGER

Cambridge University Press March 2017 (pre-order)

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SLIDE 6

Why?

Simpler!

(1⊗σ⊗k)◦(σ⊗1⊗1⊗1)◦ (f ⊗ g ⊗ 1 ⊗ 1) ◦ (h ⊗ 1) = (g ⊗ f ) ◦ (1 ⊗ k) ◦ (h ⊗ 1)

vs.

k k h f

=

f g h g

New perspective = new insights
Deriving QT from first principles ⇐ ‘diagrammatic backbone’

+ a bit of information theory e.g. Pavia 2010 and Hardy 2011 Hardy (2010): “we join the quantum picturalism revolution”

A ‘theory playground’

e.g. QT vs. real/boolean-valued/modal QT, local QT (indefinite causal structure), Spekken’s toy theory, ...

New calculational tools, applications in quantum

info/computation

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Just diagrams

Q: What kinds of behaviour can we study using just diagrams, and nothing else? A: (Non-)separability

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Separability for states

States are processes with no input:

ψ

ψ ψ ...

Interpret as: ‘preparing a system (or some systems)’

Separable states:

ψ =

ψ1 ψ2

...are boring!
Non-separable := ‘no such ψ1, ψ2 exist’, but this isn’t very

helpful

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Cup-states

‘Non-separable’ isn’t very helpful, but ‘really non-separable’ is:

Definition

A process ψ is called cup-state if there exists a process φ, called a cap-effect, such that: φ ψ = ψ φ = ψ looks like a state, but it acts like a wire.

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Cup-states

By introducing some clever notation:

:= ψ := φ

Then these equations:

φ ψ = ψ φ =

...look like this:

= =

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Yank the wire!

= =

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Quantum teleportation

Bob Aleks

ψ

quantum state

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Quantum teleportation

Bob Aleks

ψ

quantum state

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Quantum teleportation

Bob Aleks

ψ

quantum state

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Quantum teleportation

Bob Aleks

ψ

entangled state quantum state

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Quantum teleportation

Bob Aleks

ψ

entangled state quantum measurement quantum state

Ui

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Quantum teleportation

Bob

U†

i

Aleks

ψ

entangled state Bob’s fix quantum measurement quantum state

Ui

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...and it works

Bob

U†

i

Aleks

ψ

Ui

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...and it works

Bob

U†

i

Aleks

ψ

Ui

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...and it works

Bob

U†

i

Aleks

ψ

Ui

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...and it works

Bob Aleks

ψ

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If we change the process theory...

Bob

U†

i

Aleks

ψ

entangled state Bob’s fix quantum measurement quantum state

Ui

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If we change the process theory...

Bob

U†

i

Aleks

b

shared random bit do nothing OR flip bit compare bits secret bit

Ui

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If we change the process theory...

Bob

U†

i

Aleks

b

shared random bit do nothing OR flip bit compare bits secret bit

Ui

...‘classical teleportation’ is one-time-pad crypto

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...and it works

Bob

U†

i

Aleks

b

Ui

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...and it works

U†

i

Ui

Aleks Bob

b

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...and it works

b

U†

i

Bob

Ui

Aleks

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...and it works

Aleks

b

Bob

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One and the same

OTP cryptography 1882 Quantum teleportation 111 years later

Image: Hanson lab@TUDelft

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Quantum algorithms

f

N bits bit

Classical computation

⇒

f

:=

Uf

‘copy’ xor

Quantum process

⇒ simple (diagrammatic) derivations of Deutsch-Jozsa, Bernstein-Vazirani, quantum seach, and hidden subgroup algorithms.

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Graphical calculus

Quantum circuits + algebraic structure = ZX-calculus ⊕ :=

α

:=

Z(α)

:=

α

X(α)

α

... ... ...

α

...

spiders

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Graphical calculus

ZX-calculus has 4 equations:

=

α+β

... ... ...

α

... ...

β

... ... ...

α+β

... ... ... ... ... ... =

α β

π 2

π

2

π

2

π

2

π

2

=

π 2 π 2

= ... ... ... ...

...which can prove any equality between Clifford circuits (and a bit more).

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Quantum circuit simplification

π 4

π

4

π

4 π 4

π

2 π 4

= =

π

π 4

π

π

4

= ⊕ ⊕ ⊕

X π

4

S†

X π

3

⊕

⊕ ⊕ ⊕

X π

3

X

π

4

†

T †

⊕ ⊕ := :=

1

X

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Measurement-based quantum computation

Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z

α

...

π

= ...

π

=

α

... ... ... ...

α π π α

... ... ... ... ... ... ... ... ... ... ... ...

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SLIDE 35

GHZ/Mermin non-locality

quantum theory any local theory

π 2 π 2 π 2 π 2 π 2 π 2

π π π π

= =

yA

i

yC

i

zA

i

yB

i

zC

i

zB

i

yA

i

yB

i

yC

i

zC

i

zB

i

zA

i

= =

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Causal structures and indeterminism

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Quantum error correction

Stabiliser code

Zd1Zd2Zb1Zp2Zp4 Zd1Zd3Zb3Zp1Zp4 Zd2Zd3Zb2Zp3Zp4 Zb4Zp1Zp2Zp3Zp4 Xd1Xd2Xb2Xb4Xp1 Xd1Xd3Xb1Xb4Xp3 Xd2Xd3Xb3Xb4Xp2 Xb1Xb2Xb3Xb4Xp4

↔

Quantum parity check graph

↔

Encoding circuit

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Automation

Quantomatic:

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SLIDE 39

Thanks!

Picturing Quantum Processes. CUP (March 2017)
Categorical Quantum Mechanics I: Causal Quantum Processes. Coecke and Kissinger. arXiv:1510.05468
Categorical Quantum Mechanics II: Classical-Quantum Interaction. Coecke and Kissinger.

arXiv:1605.08617

Coherent Parity Check Construction for Quantum Error Correction. Chancellor, Kissinger, Zohren,
Horsman. (arXiv on Monday!)

http://quantomatic.github.io

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